Today, computer aided engineering (CAE) has been used for supporting engineers in tasks such as analysis, simulation, design, manufacture, etc. In a conventional engineering design procedure, CAE analysis (e.g., finite element analysis (FEA), finite difference analysis, meshless analysis, computational fluid dynamics (CFD) analysis, modal analysis for reducing noise-vibration-harshness (NVH), etc.) has been employed to evaluate responses (e.g., stresses, displacements, etc.). Using automobile design as an example, a particular version or design of a car is analyzed using FEA to obtain the responses due to certain loading conditions. Engineers will then try to improve the car design by modifying certain parameters or design variables (e.g., thickness of the steel shell, locations of the frames, etc.) based on specific objectives and constraints. Another FEA is conducted to reflect these changes until a “best” design has been achieved. However, this approach generally depends on knowledge of the engineers or based on a trial-or-error method.
Furthermore, as often in any engineering problems or projects, these objectives and constraints are generally in conflict and interact with one another and design variables in nonlinear manners. Thus, it is not very clear how to modify them to achieve the “best” design or trade-off. This situation becomes even more complex in a multi-discipline optimization that requires several different CAE analyses (e.g., FEA, CFD and NVH) to meet a set of conflicting objectives. To solve this problem, a systematic approach to identify the “best” design, referred to as design optimization, is used.
The optimization of such systems with more than one objective functions is referred to as multi-objective optimization. Contrary to the single-objective optimization problems (SOPs), the multi-objective optimization problems (MOPs) do not yield a single optimum solution. Instead, it results in a set of optimal solutions that represent different trade-offs among objectives. These solutions are referred to as Pareto optimal solutions or Pareto optimal solution set. Design objective function space representation of the Pareto optimal solution set is known as Pareto optimal front (POF). One of the most common strategies to find Pareto optimal solutions is to convert the multi-objective optimization problem to a single objective optimization problem and then find a single trade-off solution. There are multiple ways of converting a MOP to a SOP, namely, weighted sum strategy, inverted utility functions, goal programming, ε-constraint strategy, etc. Drawbacks from this prior art approach of converting MOP to SOP are that each optimization simulation results in a single trade-off, and multiple simulations may not end in sufficiently diverse trade-off solutions.
Recently, one of the engineering design optimization methodologies is based on genetic algorithm (GA) or evolutionary algorithm. Genetic algorithms have been demonstrated to efficiently solve multi-objective optimization problems because they result in a diverse set of trade-off solutions in a single numerical simulation. GA generally starts with randomly created populations as the initial generation. Then evolutionary schemes such as crossovers and/or mutations are used for creating future generations. One of the problems in MOEA is that Pareto optimal solutions obtained may not be diversified (i.e., spread and uniformity may be poor), or convergence maybe inadequate.
In order to improve the quality, particularly the diversity, of POF in a MOEA based engineering optimization, it would, therefore, be desirable to have improved methods and systems for achieving a set of diversified global Pareto optimal solutions in a MOEA based engineering design optimization.